About the operatior-sum representation

Suppose we have a single qubit principal system ,interacting with a single qubit environment through the transformation
U=P0[tex]\otimes[/tex]I+P1[tex]\otimes[/tex]X
where X is the usual Pauli matrix (acting on the enviornment)and P0=|0><0| ,P1=|1><1| are projectors (acting on the system ).Give the quantum operation for this process ,in the operator-sum representation ,assuming the environment starts in the state |0>

3. The attempt at a solution
I have two problem about this exercise :
1.how to represent the principal system
2.how to find an orthonormal basis for the environment so that the operatior-sum representation can be arrived